Cavity resonators are efficient, robust, electromagnetic devices that have a multitude of applications in radio-frequency, microwave, and vacuum electronics. Cavity resonators are used as frequency selective filters and oscillators. In vacuum electronic devices and charged particle accelerators cavities are used to couple energy into and out of charged particle beams. In the vast majority of these applications, only the fundamental frequency of the cavity resonator is utilized. The fundamental cavity mode generally has the greatest frequency separation from neighboring modes, has a simple field distribution, and can easily be made to be axially symmetric for interacting with an on-axis beam. The higher order modes in most cavity geometries are a mixture of transverse electric (TE) and transverse magnetic (TM) modes that increase in density with frequency until becoming a continuum. The mode spacing and fields of higher order modes are irregular and are generally undesirable in most applications. Applications that use single mode cavities are limited to sinusoidal waveforms and interactions.